You have an infinite amount of timers, each is an hour long (they do not have dials on telling you how long they've been going  they just beep when the time is up). You can set it to double speed at any time, but you cannot set it back to normal speed (eg if you set it to double speed at the start it will last 30 minutes.
Using each timer only once, is it possible to time exactly 25 minutes?
If it is, what is the smallest number of timers you need to do this, and the quickest time you can acheive it?
If you read my previous post (which BTW should say the sum from n=2 to infinity on the 1st line) I demonstrate a way to do this with an infinite amount of timers. It's impossible, though, to do this with a finite number of them.
The problem lies in the fact that 25/60=5/12=5/(3*2*2). What you're given are timers in the form 60/(2^n), where n is a positive integer. Since all these timers are power of 2 fractions of 60, it is impossible to linearly stack them to get any fraction 60 that is a multiple of 3 (since no power of 2 can be a multiple of 3). Therefore, no finite addition or subtraction of these timers can give 25 min.
The other thing that seems promising is the ability to double the speed of a timer partway through it (based on results of other timers). But upon inspection this still only amounts to adding and subtracting power of 2 fractions again. Therefore, this is impossible unless done using an infinite number of timers using the method described in my previous post.

Posted by Oren
on 20040423 16:00:50 